Why did my computer die while turned on?

What diagnostic testing should I do besides checking the PSU (first)? Beyond the PSU, I haven't a clue about what to check or how to check it. Can I get some help on this. It's an old Compaq Presario 7594 computer running Windows 98 SE. It was given to me and want to make it useable for my young grandchildren.....if it doesn't cost too much. It just went off as if it were unplugged....I turned it on and looked around a while and started reading a book, fell asleep and when I woke up, the computer was off. I tried turning it back on but it would do nothing.